In this paper, we present the first neural video codec that can compete with the latest coding standard H.266/VVC in terms of sRGB PSNR on UVG dataset for the low-latency mode. Existing neural hybrid video coding approaches rely on optical flow or Gaussian-scale flow for prediction, which cannot support fine-grained adaptation to diverse motion content. Towards more content-adaptive prediction, we propose a novel cross-scale prediction module that achieves more effective motion compensation. Specifically, on the one hand, we produce a reference feature pyramid as prediction sources, then transmit cross-scale flows that leverage the feature scale to control the precision of prediction. On the other hand, we introduce the mechanism of weighted prediction into the scenario of prediction with a single reference frame, where cross-scale weight maps are transmitted to synthesize a fine prediction result. In addition to the cross-scale prediction module, we further propose a multi-stage quantization strategy, which improves the rate-distortion performance with no extra computational penalty during inference. We show the encouraging performance of our efficient neural video codec (ENVC) on several common benchmark datasets and analyze in detail the effectiveness of every important component.